In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a series expansion is an expansion of a
function into a
series, or infinite sum.
It is a method for calculating a
function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).
The resulting so-called ''
series'' often can be limited to a finite number of terms, thus yielding an
approximation
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the
partial sum of the omitted terms) can be described by an equation involving
Big O notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund L ...
(see also
asymptotic expansion). The series expansion on an
open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
will also be an approximation for non-
analytic functions.
Types of series expansions
There are several kinds of series expansions, listed below.
A ''
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
'' is a
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
based on a function's
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s at a single point.
More specifically, if a function
is infinitely differentiable around a point
, then the Taylor series of ''f'' around this point is given by
under the convention
.
The ''
Maclaurin series
Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin ...
'' of a ''f'' is its Taylor series about
.
A ''
Laurent series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
'' is a generalization of the Taylor series, allowing negative exponent values; it takes the form
and converges in an
annulus
Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to:
Human anatomy
* ''Anulus fibrosus disci intervertebralis'', spinal structure
* Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...
.
A ''general
Dirichlet series'' is a series of the form
. One important special case of this is the ''ordinary Dirichlet series''
.
Used in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
.
A ''
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
'' is an expansion of periodic functions as a sum of many
sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
and
cosine functions.
More specifically, the Fourier series of a function
of period
is given by the expression
where the coefficients are given by the formulae
In
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acousticia ...
, e.g., the
fundamental tone and the
overtones together form an example of a Fourier series.
Newtonian series
Legendre polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applica ...
: Used in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
to describe an arbitrary electrical field as a
superposition of a
dipole field, a
quadrupole field, an
octupole field, etc.
Zernike polynomials
In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk. Named after optical physicist Frits Zernike, winner of the 1953 Nobel Prize in Physics and the inventor of phase-contrast microscopy, t ...
: Used in
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
to calculate
aberrations of optical systems. Each term in the series describes a particular type of aberration.
The ''
Stirling series''
is an approximation of the
log-gamma function.
Examples
The following is the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of
:
The Dirichlet series of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
is
References
Algebra
Polynomials
Mathematical analysis
Mathematical series